Problem: Cameron writes down the smallest positive multiple of 20 that is a perfect square, the smallest positive multiple of 20 that is a perfect cube, and all the multiples of 20 between them.  How many integers are in Cameron's list?
Answer: A perfect square that is a multiple of $20 = 2^2 \cdot 5^1$ must be a multiple of $2^2 \cdot 5^2 = 100$.  A perfect cube that is a multiple of 20 must be a multiple of $2^3 \cdot 5^3 = 1000$.  Our goal is thus to count the multiples of 20 from 100 to 1000 inclusive: $$ 100 \le 20n \le 1000. $$Dividing this entire inequality by 20 we get $5 \le n \le 50$, so there are $50 - 5 + 1 = \boxed{46}$ integers in Cameron's list.